Measurements at high angular resolution: Michelson’s stellar interferometer
 

The distances of stars are so great in relation to their diameters that, even with the world’s largest telescopes, it is extremely difficult to make measurements of their apparent diameters from conventional imaging. By extending an interferometric method which he had developed for measuring the diameters of such objects as Jupiter’s satellites, Michelson in the 1920s provided the first direct measurements of the diameters of a few stars. His interferometer was never in general use but it is worthwhile describing it here, as it represents a milestone in providing us with the basic knowledge of the sizes of stars.
Michelson’s original interferometer consisted of two collecting apertures which were fixed to a beam so that their separation could be adjusted. The length of the beam was 6 m. A very stable platform was provided for the beam by attaching it to the 100-in (2·54 m) Mt Wilson telescope. The essential elements of the optical system are depicted in figure 1. Its principle is similar to that of Young’s classical double-slit experiment.
Suppose that the interferometer is in exact adjustment so that the distances M₁, M₂, M₅, M6, eye and M₄, M₃, M₅, M₆, eye provide the same optical path length. The wavefronts arriving perpendicularly to the system are intercepted by the two apertures M₁ and M₄ and the interference pattern set up from the two rays can be viewed in the eyepiece. For the second series of wavefronts arriving at an angle θ to the first set, a second interference pattern is set up but, in general, this is slightly displaced so that it does not overlap the first. For the second set of wavefronts, the ray leaving
M₄ lags in phase behind that from M1 as it has had to travel an extra distance given by d sin θ, where d is the separation of the mirrors. If the value of d sin θ is equal to λ/2, then a bright fringe from the pattern set up by the second set of wavefronts will fall exactly on a dark fringe which was produced

Figure 1. The Michelson stellar interferometer. Mirrors M₁ and M₄ collect the light from the star and act in the same way as the slits in Young’s classical interference experiment. Wavefronts are drawn corresponding to those arriving from points separated by an angle θ.
from the first set of wavefronts. The overall effect at this condition is that the two fringe patterns are cancelled out by each other. Thus, for two point sources separated by an angle θ, a separation of the mirrors M₁ and M₄ can be found so that no interference pattern is present and, at this condition,

Hence,
(1)
When the object being investigated is a single star, the interference from each point within the optical disc needs to be considered. Mathematical procedures which are similar to those needed to evaluate the diffraction pattern produced by a circular aperture are required to determine the condition when the interference pattern is destroyed. For a circular object of uniform brightness, the interference pattern should not be detectable when
(2)
where θ now corresponds to the angular diameter of the star. If the star is known not to be uniformly bright, allowances can be made which alter the numerical factor contained in equation (2).

The original Michelson interferometer was only able to investigate a few nearby supergiant stars. One of its limitations was that the interference fringes had to be assessed by eye. The experiment requires the separation of the mirrors to be adjustable but, for the white light fringe condition, the difference in path length for the two beams needs to be maintained to the order of one wavelength of light—a difficult mechanical problem at the time. With the application of modern equipment such as lasers to monitor the mirror separation and TV cameras to record fringe patterns, there has been a resurgence of interest in the principle of Michelson interferometry by several research groups around the world.

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